Landfill leachates are a mixture of high concentration organic and inorganic contaminants and need to be appropriately treated due to their toxicity and severe adverse effects on the environment. Here, we studied the treatment of landfill leachate through a heterogeneous Fenton-like oxidation process using a zirconia supported copper catalyst (Cu/ZrO2). Reaction conditions such as pH, amount of catalyst, oxidant dose, temperature, and reaction time were investigated and their effects on pollutant abatement discussed. AOS (average oxidation state) and COS (carbon oxidation state) parameters were used for the evaluation of the degree of oxidation of the process, obtaining some insight into the formation of oxidized intermediates (partial oxidation) and the total oxidation (mineralization) of the leachate during the reaction. A two-step oxidation process enhanced the overall performance of the reaction with an abatement of organic compounds of 92% confirming the promising activity of a copper-based catalyst for the treatment of liquid waste. Higher catalytic activity was achieved when the following reaction conditions were applied: 70 °C, pH 5, 200 mg/L of catalyst, 30 mL/L of H2O2 dose, and 150 min. In addition, durability of the catalyst under optimized reaction conditions was verified by repeated reaction cycles.